International Association for Cryptologic Research

International Association
for Cryptologic Research


Counting Unpredictable Bits: A Simple PRG from One-way Functions

Noam Mazor , Tel Aviv University
Rafael Pass , Tel-Aviv University and Cornell Tech
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Presentation: Slides
Conference: TCC 2023
Abstract: A central result in the theory of Cryptography, by Hastad, Imagliazzo, Luby and Levin [SICOMP’99], demonstrates that the existence one-way functions (OWF) implies the existence of pseudo-random generators (PRGs). Despite the fundamental importance of this result, and several elegant improvements/simplifications, analyses of constructions of PRGs from OWFs remain complex (both conceptually and technically). Our goal is to provide a construction of a PRG from OWFs with a simple proof of security; we thus focus on the setting of non-uniform security (i.e., we start off with a OWF secure against non-uniform PPT, and we aim to get a PRG secure against non-uniform PPT). Our main result is a construction of a PRG from OWFs with a self-contained, simple, proof of security, relying only on the Goldreich-Levin Theorem (and the Chernoff bound). Although our main goal is simplicity, the construction, and a variant there-of, also improves the efficiency—in terms of invocations and seed lengths—of the state-of-the-art constructions due to [Haitner-Reingold-Vadhan, STOC’10] and [Vadhan-Zheng, STOC’12], by a factor O(log2 n). The key novelty in our analysis is a generalization of the Blum-Micali [FOCS’82] notion of unpredictabilty—rather than requiring that every bit in the output of a function is unpredictable, we count how many unpredictable bits a function has, and we show that any OWF on n input bits (after hashing the input and the output) has n + O(log n) unpredictable output bits. Such unpredictable bits can next be “extracted” into a pseudorandom string using standard techniques.
  title={Counting Unpredictable Bits: A Simple PRG from One-way Functions},
  author={Noam Mazor and Rafael Pass},